Projection effects in star-forming regions

Stars are formed as molecular clouds fragment into networks of dense cores, filaments, and sub-clusters. The characteristic spacing of these dense cores is therefore a key observable imprint of the underlying fragmentation physics and is often compared to theoretical scales such as the Jeans or sonic length. Nearest-neighbour (NN) statistics are widely used to measure this spacing, yet they are derived from projected 2D positions, while fragmentation unfolds in three dimensions. Using a hierarchy of spherical and fractal toy models, we show that the standard geometric de-projection factor of 4/ π ≃ 1.27 is inadequate because two effects operate together: (1) Projection not only foreshortens separations but also rewires the NN network, creating artificial 2D links between sources that are widely separated in 3D. (2) Finite angular resolution introduces beam blending, which merges close neighbours and inflates the apparent separations. We quantify these opposing biases with Monte Carlo experiments spanning a wide range of morphologies, sample sizes, and resolutions, parametrized by the number of independent beams across the field of view. From this parameter space analysis we derived a simple empirical correction factor that depends on both the number of identified objects and the effective resolution. For small samples or coarsely resolved data ( N ≲ 10 or ≲10 beams across the field), the intrinsic mean NN spacing exceeds the projected value by only ~20–40%, while for well-sampled, well-resolved maps ( N ≳ 100 and ≳30–50 beams across the field), the true 3D separations are typically larger than the observed 2D spacings by a factor of approximately two. In practice, this calibration allows observers to take a measured 2D NN spacing and estimate a corresponding 3D value by applying a resolution- and sample-size-dependent multiplicative factor, with typical morphology-driven systematic uncertainties on the order of 30–40%. We compare this framework to observed and simulated core populations and show how it modifies inferences about preferred fragmentation scales. This work is a first step towards quantifying projection bias in core separations. We deliberately omitted additional complexities such as sensitivity limits, background confusion, and incomplete field of view, and we outline paths forward via synthetic observations, hydrodynamic simulations, and velocity-resolved datasets to build a more complete framework for interpreting 2D spacing statistics in star-forming regions.